equation we have chosen to write it as a sum of a linear uniformly parabolic part and two nonlinear terms, expressed by f and g in 1.
The study of the L
p
norms w.r.t. the randomness of the space-time uniform norm on the trajecto- riesof a stochastic PDE was started by N. V. Krylov in [7]. His aim was to obtain estimates useful
for numerical approximations. In [5] we have introduced the method of iteration of Moser more precisely a version due to Aronson -Serrin for non -linear equations in the stochastic framework,
which allowed us to treat equations with measurable coefficients. The present paper is a continua- tion of these. One of our motivations is to get Holder continuity properties for the solution of the
SPDE in a forthcoming paper. As in the deterministic case we think that an essential step is to estab- lish a stochastic version of a maximum principle. Moreover, our maximum principle allows one to
estimate the solution of the Dirichlet problem with random boundary data. For simplicity, let us give a consequence of it. Under suitable assumptions on f , g, h Lipschitz continuity and integrability
conditions, we have
Theorem 2. Let M
t t
≥0
be an Itô process satisfying some integrability conditions, p ≥ 2 and u be a
local weak solution of 1. Assume that u ≤ M on the parabolic boundary {[0, T [×∂ O } ∪ {{0} × O },
then for all t ∈ [0, T ]:
E u − M
+ p
∞,∞;t
≤ k p, t E
kξ − M k
p ∞
+ f
0,M +
∗p θ ,t
+ |g
0,M
|
2 ∗p2
θ ;t
+ |h
0,M
|
2 ∗p2
θ ;t
where f
0,M
t, x = f t, x, M , 0, g
0,M
t, x = gt, x, M , 0, h
0,M
t, x = ht, x, M , 0 and k is a func- tion which only depends on the structure constants of the SPDE,
k · k
∞,∞;t
is the uniform norm on [0, t] × O and k·k
∗ θ ;t
is a certain norm which is precisely defined below. The paper is organized as follows : in section 2 we introduce notations and hypotheses and we
take care to detail the integrability conditions which are used all along the paper. In section 3 we establish Itô’s formula for the positive part of the local solution Proposition 1. In section 4, we
prove a comparison theorem Theorem 5 which yields the maximum principle Theorem 7. Then in section 5 we prove an existence result for Burgers type SPDE’s with null Dirichlet conditions and
so we generalize results obtained by Gyöngy and Rovira [6]. Moreover we shortly indicate how the maximum principle and the comparison theorem generalize to this kind of equations. Finally in the
appendix we present some technical facts related to solutions in the L
1
-sense which are used in the proofs of the preceding sections.
2 Preliminaries
2.1 L
p,q
-spaces
Let O be an open bounded domain in R
d
. The space L
2
O is the basic Hilbert space of our frame- work and we employ the usual notation for its scalar product and its norm,
u, v = Z
O
u x v x d x, kuk
2
= Z
O
u
2
x d x
1 2
. In general, we shall use the notation
u, v = Z
O
uxvx d x, 503
where u, v are measurable functions defined in O and uv ∈ L
1
O . Another Hilbert space that we use is the first order Sobolev space of functions vanishing at the
boundary, H
1
O . Its natural scalar product and norm are u, v
H
1
O
= u, v + Z
O d
X
i=1
∂
i
u x ∂
i
v x d x,
kuk
H
1
O
=
kuk
2 2
+ k∇uk
2 2
1 2
. We shall denote by H
1 l oc
O the space of functions which are locally square integrable in O and which admit first order derivatives that are also locally square integrable.
For each t 0 and for all real numbers p, q ≥ 1, we denote by L
p,q
[0, t] × O the space of classes of measurable functions u : [0, t]
× O −→ R such that kuk
p,q; t
:=
Z
t
Z
O
|us, x|
p
d x
q p
ds
1 q
is finite. The limiting cases with p or q taking the value ∞ are also considered with the use of the
essential sup norm. We identify this space, in an obvious way, with the space L
q
[0, t] ; L
p
O , con- sisting of all measurable functions u : [0, t]
→ L
p
O such that Z
t
u
s q
p
ds ∞. This identification
implies that Z
t
u
s q
p
ds
1 q
= kuk
p,q; t
. The space of measurable functions u : R
+
→ L
2
O such that kuk
2,2;t
∞, for each t ≥ 0, is denoted by L
2 l oc
R
+
; L
2
O
. Similarly, the space L
2 l oc
R
+
; H
1
O
consists of all measurable functions u : R
+
→ H
1
O such that kuk
2,2;t
+ k∇uk
2,2;t
∞, for any t
≥ 0. Next we are going to introduce some other spaces of functions of interest and to discuss a certain
duality between them. They have already been used in [1] and [5] but here intervenes a new case and we change a little bit the notation used before in a way which, we think, make things clearer.
Let p
1
, q
1
, p
2
, q
2
∈ [1, ∞]
2
be fixed and set I = I p
1
, q
1
, p
2
, q
2
:= ¦
p, q ∈ [1, ∞]
2
∃ ρ ∈ [0, 1] s.t. 1
p = ρ
1 p
1
+ 1 − ρ 1
p
2
, 1
q = ρ
1 q
1
+ 1 − ρ 1
q
2
¾ .
This means that the set of inverse pairs
1 p
,
1 q
, p, q belonging to I, is a segment contained in the square [0, 1]
2
, with the extremities
1 p
1
,
1 q
1
and
1 p
2
,
1 q
2
. There are two spaces of interest associated to I. One is the intersection space
L
I;t
= \
p,q ∈I
L
p,q
[0, t] × O .
504
Standard arguments based on Hölder’s inequality lead to the following inclusion see e.g. Lemma 2 in [5]
L
p
1
,q
1
[0, t] × O ∩ L
p
2
,q
2
[0, t] × O ⊂ L
p,q
[0, t] × O , for each p, q
∈ I, and the inequality kuk
p,q;t
≤ kuk
p
1
,q
1
;t
∨ kuk
p
2
,q
2
;t
, for any u
∈ L
p
1
,q
1
[0, t] × O ∩ L
p
2
,q
2
[0, t] × O . Therefore the space L
I;t
coincides with the inter- section of the extreme spaces,
L
I;t
= L
p
1
,q
1
[0, t] × O ∩ L
p
2
,q
2
[0, t] × O and it is a Banach space with the following norm
kuk
I;t
:= kuk
p
1
,q
1
;t
∨ kuk
p
2
,q
2
;t
. The other space of interest is the algebraic sum
L
I;t
:= X
p,q ∈I
L
p,q
[0, t] × O , which represents the vector space generated by the same family of spaces. This is a normed vector
space with the norm
kuk
I;t
:= inf
n
X
i=1
u
i p
i
,q
i
; t
u =
n
X
i=1
u
i
, u
i
∈ L
p
i
,q
i
[0, t] × O , p
i
, q
i
∈ I, i = 1, ...n; n ∈ N
∗
. Clearly one has L
I;t
⊂ L
1,1
[0, t] × O and kuk
1,1;t
≤ c kuk
I;t
, for each u ∈ L
I;t
, with a certain constant c
0. We also remark that if p, q
∈ I, then the conjugate pair p
′
, q
′
, with
1 p
+
1 p
′
=
1 q
+
1 q
′
= 1, belongs to another set, I
′
, of the same type. This set may be described by I
′
= I
′
p
1
, q
1
, p
2
, q
2
:= ½
p
′
, q
′
∃ p, q ∈ I s.t.
1 p
+ 1
p
′
= 1
q +
1 q
′
= 1 ¾
and it is not difficult to check that I
′
p
1
, q
1
, p
2
, q
2
= I
p
′ 1
, q
′ 1
, p
′ 2
, q
′ 2
, where p
′ 1
, q
′ 1
, p
′ 2
and q
′ 2
are defined by
1 p
1
+
1 p
′ 1
=
1 q
1
+
1 q
′ 1
=
1 p
2
+
1 p
′ 2
=
1 q
2
+
1 q
′ 2
= 1. Moreover, by Hölder’s inequality, it follows that one has
Z
t
Z
O
u s, x v s, x d x ds ≤ kuk
I;t
kvk
I
′
;t
, 2
for any u ∈ L
I;t
and v ∈ L
I
′
;t
. This inequality shows that the scalar product of L
2
[0, t] × O extends to a duality relation for the spaces L
I;t
and L
I
′
;t
. Now let us recall that the Sobolev inequality states that
kuk
2
∗
≤ c
S
k∇uk
2
, 505
for each u ∈ H
1
O , where c
S
0 is a constant that depends on the dimension and 2
∗
=
2d d
−2
if d
2, while 2
∗
may be any number in ]2, ∞[ if d = 2 and 2
∗
= ∞ if d = 1. Therefore one has kuk
2
∗
,2;t
≤ c
S
k∇uk
2,2;t
, for each t
≥ 0 and each u ∈ L
2 l oc
R
+
; H
1
O
. And if u ∈ L
∞ l oc
R
+
; L
2
O T
L
2 l oc
R
+
; H
1
O
, one has
kuk
2, ∞;t
∨ kuk
2
∗
,2;t
≤ c
1
kuk
2 2,
∞;t
+ k∇uk
2 2,2;t
1 2
, with c
1
= c
S
∨ 1. One particular case of interest for us in relation with this inequality is when p
1
= 2, q
1
= ∞ and p
2
= 2
∗
, q
2
= 2. If I = I 2, ∞, 2
∗
, 2 , then the corresponding set of associated conjugate numbers is I
′
= I
′
2, ∞, 2
∗
, 2 = I 2, 1,
2
∗
2
∗
−1
, 2 , where for d = 1 we make the convention that
2
∗
2
∗
−1
= 1. In this particular case we shall use the notation L
;t
:= L
I;t
and L
∗ ;t
:= L
I
′
;t
and the respective norms will be denoted by
kuk
;t
:= kuk
I;t
= kuk
2, ∞;t
∨ kuk
2
∗
,2;t
, kuk
∗ ;t
:= kuk
I
′
;t
. Thus we may write
kuk
;t
≤ c
1
kuk
2 2,
∞;t
+ k∇uk
2 2,2;t
1 2
, 3
for any u ∈ L
∞ l oc
R
+
; L
2
O T
L
2 l oc
R
+
; H
1
O
and t ≥ 0 and the duality inequality becomes
Z
t
Z
O
u s, x v s, x d x ds ≤ kuk
;t
kvk
∗ ;t
, for any u
∈ L
;t
and v ∈ L
∗ ;t
.
2.2 Hypotheses
